List of practice Questions

Find the shortest distance between the lines \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \] \text{and} \[ \vec{r_2} = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k}. \]
There are 500 students in a school, of which 230 are girls. Also, 10% of 230 girls are studying in class XII. Find the probability that a randomly chosen student is of XII class and is a girl.
Find the equation of the curve which is passing through the point \( (1, 1) \), and whose differential equation is \[ x \, dy = (2x^2 + 1) \, dx, (x \neq 0). \]
For any two vectors \( \vec{a} \) and \( \vec{b} \), prove that always \[ |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}|. \]
Evaluate: \[ \int \left( \sqrt{\cot x} + \sqrt{\tan x} \right) dx \]
If \[ \Delta = \begin{vmatrix} 3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3 \end{vmatrix} \] then find the value of $|\Delta|$.
Find the value of the expression \( i \cdot i + j \cdot j + 2k \cdot k \).
Find the degree of the differential equation \[ x y \left( \frac{d^2 y}{dx^2} \right)^2 + x \frac{dy}{dx} - y = 2. \]
Prove that \( f(x) = \tan x \) for all \( x \in \mathbb{R} \) is a continuous function.
If \( y = A \sin x + B \cos x \), then find the differential equation of it.
Find those points on the curve \[ \frac{x^2}{4} + \frac{y^2}{25} = 1, \] on which the normal is parallel to the x-axis.
If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.
Prove that \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \).
Evaluate: \[ \int \tan^4 x \sec^2 x \, dx. \]
The value of $\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)$ will be:
The value of \[ \int \frac{dx}{x^2 - a^2} \] will be:
The nature of the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y \] will be:
Prove that a one-one function \( f : \{2, 3, 4\} \to \{2, 3, 4\} \) is onto.
If \( A = \begin{bmatrix} \cos a & -\sin a \\ \sin a & \cos a \end{bmatrix} \) and \( A + A' = I \), then find the value of \( a \).
If $f: X \rightarrow Y$ is an onto function, if and only if the range of $f$ will be:
Prove that \[ \int_{0}^{\pi} \frac{x\sin x}{1+\cos^2 x}\, dx = \frac{\pi^2}{4}. \]
Minimize $Z = 6x + 3y$ under the constraints: \[ 4x + y \geq 80, x + 5y \geq 115, 3x + 2y \leq 150, x \geq 0, \; y \geq 0. \]
If the area bounded by the curve $x=y^{2}$ and the line $x=4$ is divided into two equal parts by the line $x=a$, find the value of $a$.
From a well-shuffled pack of 52 cards, three cards were drawn one after another without any replacement. What is the probability that the first two cards be King and the third be Ace?
Define symmetric and skew-symmetric matrices. If $A$ and $B$ are symmetric matrices, prove that $(AB-BA)$ is a skew-symmetric matrix.